We investigate the use of general, non-l 2 measures of data misfit and model structure in the solution of the non-linear inverse problem. Of particular interest are robust measures of data misfit, and measures of model structure which enable piecewiseconstant models to be constructed. General measures can be incorporated into traditional linearized, iterative solutions to the non-linear problem through the use of an iteratively reweighted least-squares (IRLS) algorithm. We show how such an algorithm can be used to solve the linear inverse problem when general measures of misfit and structure are considered. The magnetic stripe example of Parker (1994) is used as an illustration. This example also emphasizes the benefits of using a robust measure of misfit when outliers are present in the data. We then show how the IRLS algorithm can be used within a linearized, iterative solution to the non-linear problem. The relevant procedure contains two iterative loops which can be combined in a number of ways. We present two possibilities. The first involves a line search to determine the most appropriate value of the trade-off parameter and the complete solution, via the IRLS algorithm, of the linearized inverse problem for each value of the trade-off parameter. In the second approach, a schedule of prescribed values for the trade-off parameter is used and the iterations required by the IRLS algorithm are combined with those for the linearized, iterative inversion procedure. These two variations are then applied to the 1-D inversion of both synthetic and field time-domain electromagnetic data.
Magnetic susceptibility affects electromagnetic (EM) loop–loop observations in ways that cannot be replicated by conductive, nonsusceptible earth models. The most distinctive effects are negative in‐phase values at low frequencies. Inverting data contaminated by susceptibility effects for conductivity alone can give misleading models: the observations strongly influenced by susceptibility will be underfit, and those less strongly influenced will be overfit to compensate, leading to artifacts in the model. Simultaneous inversion for both conductivity and susceptibility enables reliable conductivity models to be constructed and can give useful information about the distribution of susceptibility in the earth. Such information complements that obtained from the inversion of static magnetic data because EM measurements are insensitive to remanent magnetization. We present an algorithm that simultaneously inverts susceptibility‐affected data for 1D conductivity and susceptibility models. The solution is obtained by minimizing an objective function comprised of a sum‐of‐squares measure of data misfit and sum‐of‐squares measures of the amounts of structure in the conductivity and susceptibility models. Positivity of the susceptibilities is enforced by including a logarithmic barrier term in the objective function. The trade‐off parameter is automatically estimated using the generalized cross validation (GCV) criterion. This enables an appropriate fit to the observations to be achieved even if good noise estimates are not available. As well as synthetic examples, we show the results of inverting airborne data sets from Australia and Heath Steele Stratmat, New Brunswick.
S U M M A R YTwo automatic ways of estimating the regularization parameter in underdetermined, minimumstructure-type solutions to non-linear inverse problems are compared: the generalized crossvalidation and L-curve criteria. Both criteria provide a means of estimating the regularization parameter when only the relative sizes of the measurement uncertainties in a set of observations are known. The criteria, which are established components of linear inverse theory, are applied to the linearized inverse problem at each iteration in a typical iterative, linearized solution to the non-linear problem. The particular inverse problem considered here is the simultaneous inversion of electromagnetic loop-loop data for 1-D models of both electrical conductivity and magnetic susceptibility. The performance of each criteria is illustrated with inversions of a variety of synthetic and field data sets. In the great majority of examples tested, both criteria successfully determined suitable values of the regularization parameter, and hence credible models of the subsurface.The inverse problem of determining a plausible spatial variation of one or more physical properties within the Earth that is consistent with a finite set of geophysical observations can be solved by formulating it as an optimization problem in which an objective function such asis minimized. The vector m contains the M parameters in the Earth model, φ d is a measure of data misfit, φ m is a measure of some property of the Earth model, and β is the regularization parameter that balances the effects of φ d and φ m . Here, the typical sum-ofsquares misfit:is considered, where d obs = (d obs 1 , . . ., d obs N ) T is the vector containing the observations, d(m) is the vector containing the data computed for the model m, and · represents the l 2 norm. It is assumed that the noise in the observations is Gaussian and uncorrelated. The weighting matrix W d is therefore the diagonal matrix W d = diag{1/σ 1 ,. . .,1/σ N }, where σ i is the standard deviation of the noise in the ith observation. It is also assumed that the relative sizes of the standard deviations are known, with only their absolute sizes unknown. That is, σ i can be expressed as σ 0σi , where theσ i (i = 1, . . . , N ) are known and the constant σ 0 is unknown. Also, the following measure of the amount of structure in the model is considered:where m ref s and m ref z are two possibly different reference models. The elements of the weighting matrices W s and W z are obtained by substituting the discretized representation of the Earth intoandrespectively, and approximating the derivative in eq. (5) by a finite difference. The coefficients α s and α z enable the appropriate balance between the two components of φ m to be achieved for a particular problem. It is also assumed that the discretization of the Earth model is sufficiently fine that the discretization does not regularize the problem. This invariable makes the discrete inverse problem of finding the parameters in the model underdetermined, thus m...
A modification of the typical minimum-structure inver-sion algorithm is presented that generates blocky, piecewise-constant earth models. Such models are often more consistent with our real or perceived knowledge of the subsurface than the fuzzy, smeared-out models produced by current minimum-structure inversions. The modified algorithm uses [Formula: see text]-type measures in the measure of model structure instead of the traditional sum-of-squares, or [Formula: see text], measure. An iteratively reweighted least-squares procedure is used to deal with the nonlinearity introduced by the non-[Formula: see text] measure. Also, and of note here, diagonal finite differences are included in the measure of model structure. This enables dipping interfaces to be formed. The modified algorithm retains the benefits of the minimum-structure style of inversion — namely, reliability, robustness, and minimal artifacts in the constructed model. Two examples are given: the 2D inversion of synthetic magnetotelluric data and the 3D inversion of gravity data from the Ovoid deposit, Voisey’s Bay, Labrador.
Seismic methods continue to receive interest for use in mineral exploration due to the much higher resolution potential of seismic data compared to the techniques traditionally used, namely gravity, magnetics, resistivity and electromagnetics. However, the complicated geology often encountered in hard-rock exploration can make data processing and interpretation difficult. Inverting seismic data jointly with a complimentary dataset can help overcome these difficulties and facilitate the construction of a common Earth model. We consider the joint inversion of seismic traveltimes and gravity data. Our joint inversion approach incorporates measures of model similarity (i.e. slowness versus density) that are both compositional and structural in nature and follow naturally from this specific data combination. We perform the inversions on unstructured grids comprised of triangular cells in 2D, or tetrahedral cells in 3D. We present our joint inversion method on a scenario inspired by the Voisey's Bay massive sulphide deposit in Labrador, Canada.
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